I recently had a new water heater installed. After installation, all of the faucets spit and ran dark for about twenty seconds and then worked as normal.

The exception is the bathroom sink faucet. The hot water spat a bit, but no hot OR cold water.... I checked the shutoffs under the sink (although no one went under there) and I checked for another shutoff between the hot water heater and the bathroom.

I'm stumped. I can't figure out why the cold water would be affected at all, and even the hot water is working at all other faucets.

I wonder if the faucet strainer got clogged with rust and crap since the water ran dark and spitty after being turned back on. Try unscrewing the strainer and cleaning the screens.

MemoryDump is probably correct. With modern faucets, the faucet itself is the only place where hot and cold meet. Strainers get plugged fairly easily. Unscrew it, flush out the debris and then put it back on - after running both hot and cold for a few minutes to flush the lines.